First try to reset your idle programming. There may not be any problem with the components. Your description sounds like you had disconnected your battery and not reset the idle program.
One more time:
1) Disconnect the negative battery terminal.
2) Turn on the headlight switch.
3) Wait at least 10 minutes.
4) Turn off headlight switch.
5) Reconnect the negative battery terminal.
6) Start engine normally (never press on the gas pedal).
7) Allow idle speed to settle for at least 15 seconds.
8) Within 70 seconds of starting the engine, apply all accessory loads the engine will see at idle. Do these concurrently.
8a) Climate control to MAX A/C.
8b) Headlight switch ON.
8c) Turn steering wheel 1/2 turn side to side.
8d) Press brake pedal.
8e) ATX SHOs only: shifter to DRIVE.
9) Do not turn off engine for at least 5 minutes.
10) It is preferable that the car be driven immediately after setting the idle programming.
11) While driving, allow the coolant temperature to rise to the normal range.
12) Run the car through its entire performance range.